Phase-Field Modeling of Cohesive Fracture. Part I: \({\Gamma }\)-Convergence Results

The main aim of this three-part work is to provide a unified consistent framework for the phase-field modeling of cohesive fracture. In this first paper we establish the mathematical foundation of a cohesive phase-field model by proving a \Gamma-convergence result in a one-dimensional setting. Specifically, we consider a broad class of phase-field energies, encompassing different models present in the literature, thereby both extending the results in [S. Conti, M. Focardi, and F. Iurlano, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 33 (2016), pp. 1033--1067] and providing an analytical validation of all the other approaches. Additionally, by modifying the functional scaling, we demonstrate that our formulation also generalizes the Ambrosio--Tortorelli approximation for brittle fracture, therefore laying the groundwork for a unified framework for variational fracture problems. The Part II paper presents a systematic procedure for constructing phase-field models that reproduce prescribed cohesive laws, whereas the Part III paper validates the theoretical results with applied examples.